On 30 Jun 2017, at 19:36, John Clark wrote:
On Fri, Jun 30, 2017 at 1:07 PM, Bruno Marchal <[email protected]>
wrote:
> It just means that there is no proof that such larger prime
number don't exist in the theory RA:
Then theory RA sucks.
It is a very useful theory, like the bacteria Escherichia Coli has
been useful for the development of molecular genetics.
You could say that E. Coli sucks because it cannot write english.
RA, which is logic +
0 ≠ s(x)
s(x) = s(y) -> x = y
x = 0 v Ey(x = s(y))
x+0 = x
x+s(y) = s(x+y)
x*0=0
x*s(y)=(x*y)+x
is the smallest finitely axiomatizable theory which is *essentially*
undecidable (which means that not only the theory is undecidable, but
all its consistent extension, (like PA, ZF, ...) are undecidable too.
If you delete one axiom above, you get an undecidable theory admitting
some decidable complete extension.
I use it because it is enough for the ontology. For the phenomenology,
I need a machine having more introspective cognitive ability. PA is
well suited by this. And PA is just RA + the induction axioms.
Note that
1) RA can (and do) emulate PA, and all other Löbian machine including
us.
It is good to keep in mind that emulating is not the same as proving
or believing. RA can prove that PA can prove RA's consistency, but
that does NOT make RA able to prove its own consistency (like I can
emulate Einstein's brain does not make me believing what Einstein will
say, cf "Searles Chinese room mistake).
2) RA is the smallest finitely axiomatizable essentially undecidable
theory, but is not the smallest Turing-complete theory. If we allow
scheme of axioms, we can find still weaker theory, like Robinson other
theory called R.
An excellent book on this (the original papers), is the Dover's book
by Tarski "Undecidable Theories" (1953, Dover 2010).
Bruno
John K Clark
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